Formally, \(\theta _{N-k(N), N}\)  is called an intermediate (or intermediate extreme) order statistic if \(k(N)\rightarrow \infty \)  as \(N\rightarrow \infty \)  and \(k(N)=o(N)\) .See chapter 2 of [1]} for a detailed discussion of asymptotic theory for intermediate order statistics in an iid setting.
Intuitively, intermediate order statistics asymptotically stay in the tail, but are not the top statistics.
We will generally suppress dependence of \(k\)  on \(N\) .
